216 
is applied for every (6£,67) inside a neighborhood yielding an overconstrained system with six unknowns. However, both 
these already known methods use assumptions about local constancy or affinity of the optical flow that do not reflect the 
harmonic variation of both flow components (3) with the angle 7 in the log-polar plane. We will delve into this problem 
in the next section. We finish this section giving abbreviations to the presented methods. As of now we will call LCT 
the method based on the Local Constancy of the flow in the Transformed image (polar or log-polar) and LAT to the 
method based on the Local Affinity of the flow in the transformed image. 
3 The polar deformation 
The first source of error in the log-polar optical flow field is due to the polar deformation of the gray-value function. 
The polar transformation maps straight edges into curved edges (see Fig. 2 (left)) enabling thus the computation of both 
components of the optical flow at points without curvature in the original cartesian image. This superficial elimination 
of the aperture problem introduces optical flow values with a large error regarding the expected motion field. In this 
section we will first transform the neighborhood-gradient approach into a second derivative method in order to study 
analytically the rank of the resulting linear system. Then we will prove the expected fact that the linear system in 
the polar plane has full rank even if the Hessian matrix in the cartesian plane is singular. The aperture problem is 
always eliminated by introducing some assumption on the local variation of the flow. In this sense, we assume the local 
constancy of the back-transformed flow in the cartesian plane. We, thus, face the aperture problem in the cartesian 
place. The resulting matrix of gray-value derivatives in the polar plane will be proved to have the same rank as the 
cartesian Hessian. We do not here consider the error due to subsampling introduced by the low-angular resolution of the 
log-polar plane. 
We denote by E(p,7n) the gray-value function and by (u?, v?) the optical flow in the polar domain. Using the 
Brightness Change Constraint Equation and the assumption that the polar flow is locally constant and applying the 
Taylor expansion to the derivatives at the positions (p + 6p, n 4- Sn) yields the overconstrained system 
Eu? + Env? + Er =0 EopuP + Enpv? + Eip = 0 EonuP + Ennv? + Ein = 0 (9) 
where we have omitted the resulting weights on each equation. We are interested in the 2 x 2 coefficient matrix of the 
second and third equation which is the Hessian of the polar gray-value function E(p, n). By twice differentiating E(p, n) 
it can be easily proved that 
  
  
  
  
Epp Enp _ 92,2)" Iz day Has 1. 5 £z dr Ÿ 3 ses - (0) 
Eo E O(p, n) Ly dy J Om) E = i E = 
with 
su) _( 85 of 
O(p, n) sz BI 
All the matrices with derivatives of (r,y) with respect to (p,m) have full rank up to two values of n for the second 
derivatives. Hence, the singularity of the cartesian Hessian does not lead to the singularity of the polar Hessian which 
also depends on the cartesian gradient. It is plausible to suppose that the smallest singular value of the coefficient matrix 
— used as a confidence measure - will be higher in the system (9) than in the equivalent system in the cartesian plane. 
This fact causes the acceptance of erroneous optical flow values. 
We proceed by substituting the assumption of local flow constancy with local flow constancy before applying the 
polar transformation. We differentiate the Brightness Change Constraint Equation assuming that the spatial derivatives 
of (u, v) vanish and we obtain a system with new coefficient matrix 
Epp Esp — d = O(z, y) Izz Izy O(z, y) 
  
  
(11) 
which proves that the singularity of the cartesian Hessian matrix is the necessary and sufficient condition for the singu- 
larity of the new coefficient matrix. Hence, the second derivative system with coefficient matrix (11) does not introduce 
erroneous values of the optical flow like the system (9). Based on this fact we are going to construct in the next section 
a gradient-sampling method like (6). 
4 Assumptions concerning the cartesian plane 
In this section we transfer the assumptions about constancy and affinity of the cartesian flow to the application of the 
BCCE in the neighborhood pixels (¢ + 8¢, 7 + 67) 
( Je(§E+ 68, n+ bn) | Jo (E -- 66,m -- 69) ) ( Ji = —J(£ + 6€, n + 6n) (12) 
allowing the log-polar flow to vary 
ul = 1 SEA d | (i 
v! poat ^t —sin(n + 6m) cos(n + 6n) v 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixeis to Sequences", Zurich, March 22-24 1995